On intrinsic negative curves

TL;DR

This paper introduces the concept of intrinsic negative curves in algebraic tori, exploring their properties and connection to Seshadri constants on toric surfaces, advancing understanding in algebraic and tropical geometry.

Contribution

It initiates a systematic study of intrinsic negative curves and links them to the computation of Seshadri constants on toric surfaces.

Findings

Intrinsic negative curves have negative self-intersection in tropical compactifications.

The study establishes a relationship between these curves and Seshadri constants.

Provides foundational insights for future research in algebraic and tropical geometry.

Abstract

Let $\mathbb K$ be an algebraically closed field of characteristic $0$. A curve of $(\mathbb K^*)^2$ arising from a Laurent polynomial in two variables is {\em intrinsic negative} if its tropical compactification has negative self-intersection. The aim of this note is to start a systematic study of these curves and to relate them with the problem of computing Seshadri constants of toric surfaces.